In considering isomorphisms of complex Banach spaces, a natural
question is whether or not real isomorphic spaces are complex
isomorphic. A negative answer was given by Bourgain and Szarek, who
exhibited, using probabilistic methods, a real Banach space which
admits two non-isomorphic complex structures.

We present a constructive version of the Bourgain-Szarek example, as
well as similar explicit constructions in contexts where the random
techniques are not suitable (as in the class of weak Hilbert spaces).

Measures corresponding to quadratic exponential families can be
characterized in a number of ways. For example, the generating
function for their orthogonal polynomials has a special form, their
Laplace transforms satisfy differential equations, etc. To
this list we add a representation on a modified symmetric Fock space.
For most of the talk, we will concentrate on the non-commutative
version of these results, which, while more abstract, are in fact
somewhat simpler.

In works of Alice Guionnet-alone, and together with Ofer
Zeitouni-have been established connections between limits of some
matrix integrals (integrals of the form INb (AN,BN) = òexp{[(Nb)/2] tr(UANU*BN)} dmNb (U), where
mNb denotes the Haar measure on the orthogonal group OAN
when b = 1 and on the unitary group UN when
b = 2, and AN,BN are diagonal real matrices), and the Burgers
equation. Roughly speaking, the Burgers equation appears in
connection to the rate function of the convergence of such integrals.
In this talk we will show how these connections can be reinterpreted
in terms of the Beltrami equation and we shall use this interpretation
to give a more complete description of the rate function. We will
apply this to a concrete example, related to the Ising model.

An abstract functorial approach to classification will be described,
including also an abstract isomorphism theorem, based on an
approximate intertwining argument, which is applicable whenever a
suitable notion of inner automorphism is present-which is decreed to
be killed by the classification functor. (Examples will be given.)

Given a unitary representation p of a locally compact group G
and a probability measure m on G, let Pm denote the
contraction Pm = òGp(g) m(dg). If X1,X2,X3...
is a sequence of i.i.d. G-valued random variables whose common
distribution is m, then the sequence p(XnXn-1 ...X1)-1Pmn converges almost surely in the strong operator
topology. This result and some of its consequences regarding a more
explicit description of the asymptotic behaviour of the powers
Pmn when n tends to ¥, will be discussed.

In 1978, Uffe Haagerup introduced an important inequality in the
context of the left-regular representation of a free group. While its
original intent was to furnish an important counter-example in the
theory of C*-algebras, Haagerup's inequality quickly found
numerous important applications in other fields: geometric group
theory, Lie theory, random walks on groups, the Baum-Connes
conjecture, and more.

The free group, or rather the von Neumann algebra generated by its
left-regular representation, is the natural arena for free
probability, a field which incorporates operator algebraic,
probabilistic, and combinatorial techniques. In this talk, I will
discuss a strengthened version of Haagerup's inequality, and a
generalization thereof to R-diagonal operators-a wide class of
non-normal operators important in free probability.

De Finetti's theorem states that an infinite exchangeable sequence of
random variables is conditionally independent. An equivalent
characterization of exchangeability was given in 1988 by Kallenberg in
terms of spreadability.

I present in my talk a new theorem that transfers these classical
results to an operator algebraic setting: An infinite
exchangeable/spreadable sequence of noncommutative random variables is
conditionally independent over its tail algebra. Here the
noncommutative version of independence is provided by Popa's commuting
squares as they are well known in subfactor theory. Surprisingly and
in contrast to the classical results, the notions of exchangeability
and spreadability are no longer equivalent. I will illustrate this
new phenomena by deformed tensor shifts on a Jones tower.

Voiculescu's theory of free probability gives a universal rule for
calculating moments of free random variables given the moments of the
individual random variables. Second order freeness does the same for
the fluctuations of random variables which are free of second order.

In this talk I will show how to extend a theorem of Krawczyk and
Speicher for cumulants of products to the second case.

Let D be the set of (non-commutative) distributions of k-tuples of
selfadjoint elements in a C*-probability space. On D one has an
operation of free additive convolution; let D¢ be the set of
distributions in D which are infinitely divisible with respect to
this operation.

In this talk I will describe a bijection B : D®D¢, obtained
in joint work with Serban Belinschi, and which is a multi-variable
version of a bijection studied by Bercovici and Pata in the case when
k=1. The bijection B is found by looking in parallel at two
"transforms" for non-commutative distributions, the eta-series and
the R-transform. We prove a theorem of convergence in moments which
parallels the Bercovici-Pata result from the case k=1. On the
other hand we prove that, quite surprisingly, B is a homomorphism
for the operation of free multiplicative convolution. An
interpretation of this fact is that eta-series share the nice
behaviour which R-transforms were known to have in connection to the
multiplication of free k-tuples of non-commutative random
variables.

The Urysohn universal metric space U is a remarkable
object, which can be described (Vershik) as the completion of the
integers equipped with a random (or: generic) metric. In many
regards, it is similar to the unit sphere S¥ of a
separable Hilbert space l2. There are however some properties
long since established for the unit sphere (e.g. the
distortion property) that remain open for the Urysohn space, and vice
versa. We will discuss an example of the latter: Connes' Embedding
Conjecture, whose analogue for the Urysohn space has been recently
settled.

As a consequence of Kirchberg's work, Connes' Conjecture can be
reformulated as follows: every pair of commuting subgroups of the
unitary group U(l2) can be approximated with pairs of commuting
compact subgroups. In this form, the property (which we call
Kirchberg property) makes sense for every topological group admitting
a chain of compact subgroups with dense union. Even if such groups
are very common among "infinite-dimensional" groups (the infinite
symmetric group, the groups of measure and measure class preserving
automorphisms, etc.), it seems the Kirchberg property has never
been verified for any concrete example. In a recent joint work
with V. V. Uspenskij, we have established the Kirchberg property for
the group of isometries of the universal Urysohn metric
space U.

We explain the relevance of the classical Riccati map in describing
the spectral edge for various random Schroedinger operators in finite
volume as well as for certain random matrix ensembles. In the latter
case these ideas provide a new characterization of the celebrated
Tracy-Widom laws of RMT in terms of the explosion probability of a
given one-dimensional diffusion.

This represents joint work with both Balint Virag (Univ. Toronto) and
Jose Ramirez (Univ. Costa Rica).

The study of weighted ergodic theorems and of special operators in
ergodic theory has led to many new insights and advances. While
harmonic analysis is often an important tool in this study,
probabilistic constructions also play a significant role. Examples of
the use of probabilistic constructions in ergodic theory will be
described.

I present a factoring algorithm that factors N=UV provably in
O(N1/3+e) time. I also discuss the potential for
improving this to a sub-exponential algorithm. Along the way, I
consider the distribution of solutions (x,y) to xy=Nmoda.

Let G be a locally compact group, and let m be a probability
measure on G. Then a function fÎL¥(G) is said to be
m-harmonic if m*f = f. The m-harmonic
functions do not form a von Neumann subalgebra of L¥(G), but
can be equipped with a product turning them into a von Neumann algebra
in its own right. Dual to this situation, for a continuous, positive
definite function s on G with s(1) = 1, A. T.-M. Lau
and C.-H. Chu called an element x of the group von Neumann algebra
VN(G) of G, s-harmonic if s·x = x. Interestingly, the collection of all s-harmonic elements
is a von Neumann subalgebra of VN(G).

Recently, W. Jaworski and M. Neufang extended the notion of a harmonic
function to that of a harmonic operators. In this talk, which is
based on joint work with Neufang, we develop a theory of harmonic
operators from the dual perspective, thus extending Lau's and Chu's
approach.

Operator theory studies elements of B(H); what happens when B(H)
is replaced with a different von Neumann algebra? Some theorems still
work, others do not, and both outcomes can reveal interesting
phenomena. In this talk I will discuss results concerning approximate
equivalence, essential spectra, and subnormal operators. The
springboard is a simple description, in terms of spectral measures, of
the norm and strong* closures of the unitary orbit of a normal
operator in a von Neumann algebra.

Grigorchuk and \.Zuk used a representation of the lamplighter group
as a self-similar group to compute the spectral measure of a simple
random walk on this group. They also introduced the notion of the
Kesten-von Neumann-Serre (KNS) spectral measure for a self-similar
group and gave several examples of the computation of such spectral
measures.

In joint work with Kambites and Silva, we characterized when the (KNS)
spectral measure coincides with the usual spectral measure; in light
of our results and an unpublished result of Abert, this coincidence of
measures will occur for any self-similar action of a non-elementary
hyperbolic group and for any bireversible automaton group. Using a
self-similar representation constructed earlier by the speaker and
Silva, we computed the spectral measure for a simple random walk on a
wreath product G\wr Z, where G is any finite group. This
same result was obtained by Dicks and Schick via a different method.

According to a classical theorem, the sequence a0,a1,a2,... is
the moment sequence of a random variable if and only if the infinite
matrix Ai,j = ai+j is positive semi-definite. We study a
graph theoretic analogue of this theorem.